2.3.1 Modified Smith Predictor
As mentioned in the previous section, the classical Smith predictor cannot be applied to unstable time-delay systems. It is not a problem to design a controllerC to stabilise the delay-free part of the system; the problem is that the predictorZ defined in (2.4) is unstable. If it is possible to find a stableZ, then the Smith predictor may still be of use.
2.3 Modified Smith Predictor (MSP)-based Control 31 Assume that the delay-free partP has a state-space realization
P= A B
C 0
and denote
Pˆ =
A B Ce−Ah 0
, then
Z= ˆP−P e−sh=Ce−Ah
h
0 e−(sI−A)ζdζB (2.7) is stable. This is known as the modified Smith predictor [147]. If this Z is plugged into Figure 2.4, then the scheme is applicable to unstable systems provided thatZis implemented as one stable block. Note that there might be unstable pole–zero cancellations insideZ ifP is unstable. Because of this,Z cannot be implemented as the difference ofPˆ andP e−sh and its implemen- tation is not trivial; see Part II for more details.
Plug (2.7) into Figure 2.4, assuming there is no modelling error, then Tyd= P e−sh
1 +1+CZC P e−sh = P(1 +CZ)e−sh 1 +CZ+CP e−sh
= CP
1 +CPˆZe−sh+ P
1 +CPˆe−sh, and
Tyr =
C
1+CZP e−sh 1 + 1+CZC P e−sh
= CP
1 +CPˆe−sh.
These equations are very similar to those obtained with the classical SP.
What’s different is that C should be designed to stabilise Pˆ so that 1+CCPˆ
P
satisfies the given specification.
The MSP can be applied to some stable systems as well, although with some caution. See Chapter 10. In this case, the open-loop poles, which dom- inate the disturbance response in the SP case, can be removed and the MSP can be implemented as the difference ofPˆ andP e−sh.
2.3.2 Zero Static Error
Since, in general,P(0)= ˆP(0), incorporating an integrator inC(s) does not guarantee zero static error for step reference tracking or disturbance rejection.
In order to recover this property, it is necessary to guarantee that the predictor Z has a zero static gain. Adding the constant−Ch
0 e−AζdζBtoPˆ results in
Pˆ=
A B Ce−Ah 0
−C
h
0 e−AζdζB, (2.8)
Z =Ce−Ah
h
0 e−(sI−A)ζdζB−C
h
0 e−AζdζB. (2.9) The above Pˆ and Z guarantees Pˆ(0) = P(0) and Z(0) = 0 and the static error is zero for step references and disturbances.
2.3.3 Simulation Examples
Two examples are given: one is a stable system to show that the dominant dynamics in the disturbance response can be removed, and the other is an unstable system to show the effectiveness of MSP.
Example 1
Consider the plant G(s) = s+11 e−0.1s studied in Subsection 2.1.3. Here, the controller in Figure 2.4 incorporates the MSP
Z = e0.1
s+ 1−(e0.1−1)− 1
s+ 1e−0.1s. The main controller is chosen to beC(s) =Kp(1 + T1
is)to stabilise Pˆ= e0.1
s+ 1−(e0.1−1) = (1−e0.1)s+ 1 s+ 1 . As a result,
CP
1+CPˆ = Kp(Tis+ 1)
Tis(s+ 1) +Kp(Tis+ 1)((1−e0.1)s+ 1). (2.10) From this, the stability condition (for positive parameters) is found to be
0< Kp< 1
e0.1−1, Ti> Kp
Kp+ 1(e0.1−1). This is shown as the shaded area in Figure 2.10.
Case 1: Ti≥1
The root locus of the system KTp
i
Tis+1 s
(1−e0.1)s+1
s+1 has the form shown in Figure 2.11(a). The zeros=−1/Tilies betweens= 0ands=−1. Hence, the two closed-loop poles are all real and one of them is between0and1. This pole dominates the system response and, hence, the closed-loop response is slower than the open-loop response. This can be verified from the system responses shown in Figure 2.12 forKp= 7andTi= 2, where a step disturbanced=−0.5
2.3 Modified Smith Predictor (MSP)-based Control 33
0 1 2 3 4 5 6 7 8 9 10
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Ti
Kp
Figure 2.10.MSP Example 1: Stability region of(Kp, Ti)
−2 0 2 4 6 8 10 12
−0.04
−0.02 0 0.02 0.04
Imaginary Axis
Real Axis
(a) Case 1:Ti= 2
−20 −15 −10 −5 0 5 10 15
−10
−5 0 5 10
Imaginary Axis
Real Axis
(b) Case 2:Ti= 0.2
Figure 2.11. Root-locus design of PI controller for MSP-based stable system
was applied att= 5s. The dominant time constant is about2.1s. Although robustness is very good, the system response is too slow.
Case 2: KKp
p+1(e0.1−1)< Ti<1
In this case, the root locus of the system KTp
i
Tis+1 s
(1−e0.1)s+1
s+1 has the form shown in Figure 2.11(b). The zeros=−1/Tilies on the left of the poles=−1.
0 1 2 3 4 5 6 7 8 9 10 0
0.2 0.4 0.6 0.8 1 1.2
Time (Seconds)
y
(a) Nominal response
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Time (Seconds)
y
(b) Robust response whenhincreases by80%
Figure 2.12.Performance of the MSP-based stable system:Kp= 7,Ti= 2
Bearing in mind that the closed-loop system bandwidth is limited to around
2hπ = 15.7 rad/s, the control parameters are then chosen asKp= 7, Ti= 0.2 to provide a damping ratio of 0.709and a pair of complex closed-loop poles
−8.15±j8.1. The system responses are shown in Figure 2.13, where a step disturbanced=−0.5 was applied att= 5 s. It can be seen that the system response is very fast and the open-loop dynamics (with a time constant of1) no longer exists in the system response. The set-point response seems undesirable, but the large overshoot is due to the term Tis+ 1 in the numerator of the response (2.10). The overshoot can be considerably reduced by adding a low- pass filter T 1
is+1 to the set-point, which offers a 2-degree-of-freedom control structure. See Figure 2.13(c) for the corresponding nominal response. The5%
overshoot is now consistent with the damping ratio of0.709.
2.3 Modified Smith Predictor (MSP)-based Control 35
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Time (Seconds)
y
(a) Nominal response
0 1 2 3 4 5 6 7 8 9 10
0 0.5 1 1.5 2
Time (Seconds)
y
(b) Robust response whenhincreases by40%
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2
Time (Seconds)
y
(c) Nominal response with a low-pass filter
Figure 2.13. Performance of the MSP-based stable system:Kp= 7,Ti= 0.2
Example 2
Consider the following unstable system:
G(s) = 1 s−1e−sh. According to (2.8, 2.9), then
Pˆ = e−h
s−1 +e−h−1 = (e−h−1)s+ 1 s−1 , Z= e−h
s−1 +e−h−1− 1 s−1e−sh. The main controller is chosen to beC(s) =Kp(1 + T1
is). As a result,
CP
1+CPˆ = Kp(Tis+ 1)
Tis(s−1) +Kp(Tis+ 1)((e−h−1)s+ 1). (2.11) It can be found that the stability condition (for positive parameters) is5
1< Kp< 1
1−e−h, Ti> Kp
Kp−1(1−e−h).
1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 0
5 10 15 20 25 30 35 40
T i
Kp
Figure 2.14.MSP Example 2: Stability region of(Kp, Ti)
This is the shaded area shown in Figure 2.14 forh= 2(the controller will be designed for h= 2 s). The admissible gain has a very limited range: from 1
5 The first condition also guaranteesC(∞) ˆP(∞)<1.
2.3 Modified Smith Predictor (MSP)-based Control 37 to 1−1e−h. The longer the delayh, the narrower the gain range. Moreover, the integral time has to be large enough as well. The longer the delayh, the larger the integral timeTi. This is because the delay and the unstable pole all impose fundamental limitations on the achievable performances [4, 17, 116, 128].
It is very tricky when tuning the parameters. Only whenTiis large enough, it is possible to find a stabilising controller. ForTi= 50, the root locus is shown in Figure 2.15, from whichKp is chosen asKp = 1.08 to obtain a damping ratio of0.828and a pair of complex closed-loop poles−0.475±j0.322. This offers an overshoot of about1%.
−1 −0.5 0 0.5 1 1.5
−0.5 0 0.5
ImaginaryAxis
Real Axis
Figure 2.15.Root-locus design for MSP-based unstable system
The system responses are shown in Figure 2.16, where a step disturbance d = −0.1 was applied at t = 20 s and a filter T 1
is+1 was added to the set- point. Figure 2.16(a) shows the nominal responses: the unstable one, denoted as “difference”, obtained when Z is implemented as the difference of Pˆ and P e−shand the stable one obtained whenZ is implemented as
Z=e−h−1 +e−hãeNh−e−Nhs eNh−1
eNh −1
s/+ 1ãΣNi=0−1e−iNh(s−1),
with N = 40 and = 0.01. See Chapter 11 for more details about this im- plementation. There is a very big dynamic error in the disturbance response.
This is because the unstable pole works in open-loop for a period of h= 2s, only after which the feedback controller starts regulating the disturbance. So the minimal maximum dynamic error is
dãeh=−0.1ìe2=−0.74.
If the control parameters are chosen as Kp = 1.112and Ti = 20, then the system is faster and the maximum dynamic error is close to the minimum one
−0.74; see Figure 2.16(b). However, the system is not robustly stable even for h = 2.05 s. The slower one with Ti = 50 is robustly stable for h = 2.05 s;
see Figure 2.16(c). The system is very easy to destabilise by modelling errors, because of the long delay and the unstable pole.
0 5 10 15 20 25 30 35 40
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (Seconds)
y
difference approximate
(a) Nominal response: different implementations
0 5 10 15 20 25 30 35 40
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (Seconds)
y
Kp=1.08, T i=50 Kp=1.112, T
i=20
(a) Nominal response: different parameters
0 20 40 60 80 100
−0.5 0 0.5 1 1.5 2
Time (Seconds)
y
(b) Robust response:hincreases toh= 2.05s (Kp= 1.08,Ti= 50) Figure 2.16. Performance of the MSP-based unstable system
2.5 Connection Between MSP and FSA 39